The Variation of Harmonic Differentials and their Periods

Abstract

The purpose of the present paper is to construct harmonic and holomorphic differentials with suitably prescribed periods on a compact Riemann surface of genus g. We use these constructions to investigate the variation of the period matrices along a curve W tµ in Teichmuller space given by a linear family tµ of Beltrami differentials. In the case of a Teichmuller geodesic, i.e., when tµ is a Teichmuller differential \(t\bar{Q}/|Q|\), we obtain some convexity properties for the real period matrix. The formulae we derive for the variation of the harmonic differential with given periods and for the period matrix are exact. We thus have the variations of all orders, and it is by looking at the second variation that we obtain our convexity result.